Lecture 3

Temporal Models and Smoothing

Sara Martino
sara.martino@ntnu.no

Dept. of Mathematical Science, NTNU

Janine Illian
Janine.Illian@glasgow.ac.uk

University of Glasgow

Motivation

Data are often observed in time, and time dependence is often expected.

  • Observations are correlated in time
  • We also have correlations between the time series (will look at the later…)

Motivation

  1. Smoothing of the time effect

What is our goal?

  1. Smoothing of the time effect

Note: We can use the same model to smooth covariate effects!

What is our goal?

  1. Smoothing of the time effect

  2. Prediction

We can “predict” any unobserved data, does not have to be in the future

Modeling time with INLA

Time can be indexed over a

  • Discrete domain (e.g., years)

  • Continuous domain

Modeling time with INLA

Time can be indexed over a

  • Discrete domain (e.g., years)

    • Main models: RW1, RW2 and AR1

    • NB RW1 and RW2 are also used for smoothing

  • Continuous domain

    • Here we use the so-called SPDE-approach

Discrete time modelling

Example - (log) Number of Air Passengers in time

Goal we want understand the pattern and predict into the future

Random Walk models

Random walk models encourage the mean of the linear predictor to vary gradually over time.

They do this by assuming that, on average, the time effect at each point is the mean of the effect at the neighboring points.

  • Random Walk of order 1 (RW1) we take the two nearest neighbors

  • Random Walk of order 2 (RW2) we take the four nearest neighbors

Random walks of order 1

Definition

\[ \begin{aligned} \pi(\mathbf{u} \mid \sigma^2) & \;\propto\; \exp\!\left( -\frac{1}{2\sigma^2} \sum_{t=1}^{T-1} (u_{t+1} - u_t)^2 \right)\\ & \;=\; \exp\!\left( -\frac{1}{2\sigma^2} \sum_{t \sim t'} ({u_t - u_{t'}})^2 \right) = \exp\!\left(-\tfrac{1}{2} \, \mathbf{u}^{\top} \mathbf{Q}\ \mathbf{u}\right) \end{aligned} \] where \(t \sim t'\) indicates \(t\) is a neighbor of \(t'\), and the precision is \(\mathbf{Q} = \mathbf{R}/\sigma^2\) with

\[ \mathbf{R} = \begin{bmatrix} 1 & -1 & & & & \\ -1 & 2 & -1 & & & \\ & -1 & 2 & -1 & & \\ & & \ddots & \ddots & \ddots & \\ & & & -1 & 2 & -1 \\ & & & & -1 & 1 \end{bmatrix} \]

What is the role of the precision parameter?